MNRAS 480, 4025–4039 (2018) doi:10.1093/mnras/sty2132Advance Access publication 2018 August 6

Short-lived radioisotopes in meteorites from Galactic-scale correlated starformation

Yusuke Fujimoto,1‹ Mark R. Krumholz1 and Shogo Tachibana2

1Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia2UTokyo Organization for Planetary and Space Science (UTOPS), The University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan

Accepted 2018 August 3. Received 2018 August 2; in original form 2018 February 18

ABSTRACTMeteoritic evidence shows that the Solar system at birth contained significant quantities ofshort-lived radioisotopes (SLRs) such as 60Fe and 26Al (with half-lives of 2.6 and 0.7 Myr,respectively) produced in supernova (SN) explosions and in the Wolf–Rayet winds that precedethem. Proposed explanations for the high SLR abundance include formation of the Sun in anSN-triggered collapse or in a giant molecular cloud (GMC) that was massive enough to survivemultiple SNe and confine their ejecta. However, the former scenario is possible only if theSun is a rare outlier among massive stars, while the latter appears to be inconsistent with theobservation that 26Al is distributed with a scale height significantly larger than GMCs. In thispaper, we present a high-resolution chemohydrodynamical simulation of the entire Milky-Way Galaxy, including stochastic star formation, H II regions, SNe, and element injection, thatallows us to measure for the distribution of 60Fe/56Fe and 26Al/27Al ratios over all stars in theGalaxy. We show that the Solar system’s abundance ratios are well within the normal range,but that SLRs originate neither from triggering nor from confinement in long-lived cloudsas previously conjectured. Instead, we find that SLRs are abundant in newborn stars becausestar formation is correlated on galactic scales, so that ejecta preferentially enrich atomic gasthat will subsequently be accreted onto existing GMCs or will form new ones. Thus, newgenerations of stars preferentially form in patches of the Galaxy contaminated by previousgenerations of stellar winds and SNe.

Key words: hydrodynamics – methods: numerical – meteorites, meteors, meteoroids – ISM:kinematics and dynamics – Galaxy: disc – gamma-rays: ISM.

1 IN T RO D U C T I O N

Short-lived radioisotopes (SLRs) – 10Be, 26Al, 36Cl, 41Ca, 53Mn,60Fe, 107Pd, 129I, 182Hf, and 244Pu – are radioactive elements withhalf-lives ranging from 0.1 Myr to more than 15 Myr that existed inthe early Solar system (e.g. Adams 2010). They were incorporatedinto meteorites’ primitive components such as calcium–aluminum-rich inclusions (CAIs), which are the oldest solids in the Solarprotoplanetary disc, or chondrules, which formed ∼1 Myr afterCAI formation. The radioactive decay of these SLRs fundamentallyshaped the thermal history and interior structure of planetesimals inthe early Solar system, and thus is of central importance for core-accretion planet formation models. The SLRs, particularly 26Al,were the main heating sources for the earliest planetesimals andplanetary embryos from which terrestrial planets formed (Grimm& McSween 1993; Michel, DeMeo & Bottke 2015), and are re-sponsible for the differentiation of the parent bodies of magmatic

� E-mail: yusuke.fujimoto@anu.edu.au

meteorites in the first few Myr of the Solar system (Greenwood et al.2005; Schersten et al. 2006; Sahijpal, Soni & Gupta 2007). TheSLRs are, moreover, potential high-precision and high-resolutionchronometers for the formation events of our Solar system due totheir short half-lives (Kita et al. 2005; Krot et al. 2008; Amelin et al.2010; Bouvier & Wadhwa 2010; Connelly et al. 2012).

Detailed analyses of meteorites show that the early Solar sys-tem contained significant quantities of SLRs. The presence of 26Alin the early Solar system was first identified in CAIs from theprimitive meteorite Allende in 1976, defining a canonical initial26Al/27Al ratio of ∼5 × 10−5 (Lee, Papanastassiou & Wasser-burg 1976, 1977; Jacobsen et al. 2008), far higher than the ratio of26Al/27Al in the interstellar medium (ISM) as estimated from contin-uous Galactic nucleosynthesis models (Meyer & Clayton 2000) andγ -ray observations measuring the in- itu decay of 26Al (Diehl et al.2006).

Compared to 26Al/27Al, the initial ratio of 60Fe/56Fe is still some-what uncertain; analyses of bulk samples of different meteoritetypes produced a low initial ratio of 60Fe/56Fe ∼1.15 × 10−8 (Tang

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4026 Y. Fujimoto, M. R. Krumholz and S. Tachibana

& Dauphas 2012, 2015), while other studies of chondrules us-ing in situ measurements found higher initial ratio of 60Fe/56Fe∼5−13× 10−7 than the ISM ratio (e.g. Mishra & Goswami 2014).Telus et al. (2016) found that the bulk sample estimates were skewedtoward low initial 60Fe/56Fe ratios because of fluid transport of Feand Ni during aqueous alteration on the parent body and/or duringterrestrial weathering, and Telus et al. (2018) have found the initialratios of 60Fe/56Fe as high as ∼0.85−5.1× 10−7, although the initial60Fe/56Fe value is still a matter of debate. If estimates in the middleor high end of the plausible range prove to be correct, they wouldimply a 60Fe/56Fe ratio well above the interstellar average as well.

It has been long debated how the early Solar system came tohave SLR abundances well above the ISM average. The isotopes26Al and 60Fe, on which we focus in this paper, are of particularinterest because they are synthesized only in the late stages ofmassive stellar evolution, followed by injection into the ISM bystellar winds and supernovae (SNe, Huss et al. 2009). Other SLRs(e.g. 10Be, 36Cl, and 41Ca) can be produced in situ by irradiation ofthe protoplanetary disc by the young Sun (Heymann & Dziczkaniec1976; Shu, Shang & Lee 1996; Lee et al. 1998; Shu et al. 2001;Gounelle et al. 2006).1 Explaining the origin site of the 26Al and60Fe, and how they travelled from this site to the primitive Solarsystem before decaying, is an outstanding problem.

One possible origin site is asymptotic giant branch (AGB) stars(Wasserburg et al. 1994; Busso, Gallino & Wasserburg 1999;Wasserburg et al. 2006). However, because AGB stars only provideSLRs at the end of their lives, and because their main-sequencelifetimes are long (>1 Gyr), the probability of a chance encounterbetween an AGB star and a star-forming region is very low (Kast-ner & Myers 1994). For these reasons, the SNe and stellar windsof massive stars, which yield SLRs much more quickly after starformation, are thought to be the most likely origin of 26Al and 60Fe.Proposed mechanisms by which massive stars could enrich the in-fant Solar system fall into three broad scenarios: (1) SN-triggeredcollapse of pre-solar dense cloud core, (2) direct pollution of analready-formed proto-solar disc by SN ejecta, and (3) sequentialstar formation events in a molecular cloud.

The first scenario, SN-triggered collapse of pre-solar dense cloudcore, was proposed by Cameron & Truran (1977) just after the firstdiscovery of 26Al in Allende CAIs by Lee et al. (1976). In thisscenario, a nearby Type II SN injects SLRs and triggers the collapseof the early Solar nebula. Many authors have simulated this scenario(Boss 1995; Foster & Boss 1996; Boss et al. 2010; Gritschnederet al. 2012; Li, Frank & Blackman 2014; Boss & Keiser 2014;Boss 2017) and shown that it is in principle possible. A single SNshock that encounters an isolated marginally stable prestellar corecan compress it and trigger gravitational collapse while at the sametime generating Rayleigh–Taylor instabilities at the surface that mixSLRs into the collapsing gas. However, these simulations have alsodemonstrated that this scenario requires severe fine-tuning. If theshock is too fast then it shreds and disperses the core rather thantriggering collapse, and if it is too slow then mixing of SLRs doesnot occur fast enough to enrich the gas before collapse. Only a verynarrow range of shock speeds are consistent with what we observein the Solar system, and even then the SLR injection efficiency islow (Gritschneder et al. 2012; Boss & Keiser 2014; Boss 2017).A possible solution to overcome the mixing barrier problem is theinjection of SLRs via dust grains. However, only grains with radii

1Small amounts of 26Al can also be produced by this mechanism, but muchtoo little to explain the observed 26Al/27Al ratio (Duprat & Tatischeff 2007).

larger than 30 μm, which is much larger than the typical sizes ofSN grains (<1 μm), can penetrate the shock front and inject SLRsinto the core (Boss & Keiser 2010). Furthermore, analysis of Aland Fe dust grains in SN ejecta constrains their sizes to be less than0.01 μm (Bocchio et al. 2016). Dwarkadas et al. (2017) proposeda triggered star formation inside the shell of a Wolf–Rayet (WR)bubble, and found that the probability is from 0.01 to 0.16.

The second scenario is a direct pollution: the Solar system’s SLRswere injected directly into an already-formed protoplanetary discby SN ejecta within the same star-forming region (Chevalier 2000;Hester et al. 2004). Hydrodynamical simulations of a protoplanetarydisc have shown that the edge-on disc can survive the impact of anSN blast wave, but that in this scenario only a tiny fraction of the SNejecta that strike the disc are captured and thus available to explainthe SLRs we observe (Ouellette, Desch & Hester 2007; Close &Pittard 2017). Ouellette et al. (2007) suggest that dust grains mightbe a more efficient mechanism for injecting SLRs into the disc,and simulations by Ouellette, Desch & Hester (2010) show thatabout 70 per cent of material in grains larger than 0.4 μm can becaptured by a protoplanetary disc. However, extreme fine-tuning isstill required to make this scenario work quantitatively. One canexplain the observed SLR abundances only if SN ejecta are clumpy,the Solar nebula was struck by a clump that was unusually rich in26Al and 60Fe, and the bulk of these elements had condensed intolarge dust grains before reaching the Solar system. The probabilitythat all these conditions are met is very low, 10−3–10−2. Moreover,the required dust size of 0.4 μm is still a factor of 40 larger than thevalue of 0.01 μm obtained by detailed study of dust grain propertiesby Bocchio et al. (2016).

The third scenario is sequential star formation events and self-enrichment in a giant molecular cloud (GMC, Gounelle et al. 2009;Gaidos et al. 2009; Gounelle & Meynet 2012; Young 2014, 2016).Gounelle & Meynet (2012) proposed a detailed picture of this sce-nario; in a first star formation event, SNe from massive stars inject60Fe to the GMC, and the shock waves trigger a second star forma-tion event. This second star formation event also contains massivestars, and the stellar winds inject 26Al and collect ISM gas to builda dense shell surrounding an H II region. In the already enricheddense shell, a third star formation event occurs where the Solar sys-tem forms. Vasileiadis, Nordlund & Bizzarro (2013) and Kuffmeieret al. (2016) have modelled the evolution of a GMC by hydro-dynamical simulations and shown that SN ejecta trapped within aGMC can enrich the GMC gas to abundance ratios of 26Al/27Al∼10−6–10−4 and 60Fe/56Fe ∼10−7–10−5, comparable to or higherthan any meteoritic estimates. However, this scenario requires thatthe bulk of the SLRs that are produced be captured within theirparent GMCs. This is enforced by fiat in the simulations (by theuse of periodic boundary conditions), but it is far from clear if thisrequirement can be met in reality. In the simulations, the requiredenrichment levels are not reached for ∼15 Myr, but observed youngstar clusters are always cleared of gas by ages of �5 Myr (e.g.Hollyhead et al. 2015). Moreover, the observed distribution of 26Alhas a scale height significantly larger than that of GMCs, whichwould seem hard to reconcile with the idea that most 26Al remainsconfined to the GMC where it was produced (Bouchet, Jourdain &Roques 2015).

The literature contains a number of other proposals (e.g. Tatis-cheff, Duprat & de Sereville 2010; Goodson et al. 2016), but whatthey have in common with the three primary scenarios outlinedabove is that they require an unusual and improbable conjunctionof circumstances (e.g. a randomly passing WR star, SN-producedgrains much larger than observations suggest) that would render

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the Solar system an unusual outlier in its abundances, or that theyare not consistent with the observed distribution of 26Al in theGalaxy.

Here, we present an alternative scenario, motivated by two ob-servations. First, 26Al is observed to extend to a significant heightabove and below the Galactic disc, suggesting that regions contam-inated by SLRs much be at least kpc scale (Bouchet et al. 2015).Second, there is no a priori reason why one should expect starformation to produce a SLR distribution with the same mean asthe ISM as a whole, because star formation does not sample fromthe ISM at random. Instead, star formation and SLR production areboth highly correlated in space and time (e.g. Efremov & Elmegreen1998; Gouliermis et al. 2010, 2015, 2017; Grasha et al. 2017a,b);the properties of GMCs are also correlated on galactic scales (e.g.Fujimoto et al. 2014, 2016; Colombo et al. 2014). That both SLRsand star formation are correlated on kpc scales suggests that it isat these scales that we should search for a solution to the origin ofSLRs in the early Solar system.

In this paper, we will study the Galactic-scale distributions of26Al and 60Fe produced in stellar winds and SNe, and propose anew contamination scenario: contamination due to Galactic-scalecorrelated star formation. In Section 2, we present our numericalmodel of a Milky-Way-like galaxy, along with our treatments ofstar formation and stellar feedback. In Section 3, we describe globalevolution of the galactic disc and the abundance ratios of the starsthat form in it. In Section 4, we discuss the implications of ourresults, and based on them we propose a new scenario for SLRdeposition. We summarize our findings in Section 5.

2 ME T H O D S

We study the abundances of 60Fe and 26Al in newly formed stars byperforming a high-resolution chemohydrodynamical simulation ofthe ISM of a Milky-Way-like galaxy. The simulation includes hy-drodynamics, self-gravity, radiative cooling, photoelectric heating,stellar feedback in the form of photoionization, stellar winds, andSNe to represent dynamical evolution of the turbulent multiphaseISM, and a fixed axisymmetric logarithmic potential to represent thegravity of old stars and dark matter, which causes the galactic-scaleshear motion of the ISM in a flat rotation curve. In the simula-tion, when self-gravity causes the gas to collapse past our ability toresolve, we insert ‘star particles’ that represent stochastically gen-erated stellar populations drawn star-by-star from the initial massfunction (IMF). Each massive star in these populations evolves indi-vidually until it produces a mass-dependent yield of 60Fe and 26Al atthe end of its life. We subsequently track the transport and decay ofthese isotopes, and their incorporation into new stars. Further detailson our numerical method are given in the following subsections.

We carry out all analysis and post-processing of the simulationoutputs, and produce all simulations visualizations, using the YT

software package (Turk et al. 2011).

2.1 Chemohydrodynamical simulation

Our simulations follow the evolution of a Milky-Way-type galaxyusing the adaptive mesh refinement code ENZO (Bryan et al. 2014).We use a piecewise parabolic mesh hydrodynamics solver to followthe motion of the gas. Since the ∼200 km s−1 circular velocity ofthe galaxy necessitates strongly supersonic flows in the galacticdisc, we make use of the dual energy formalism implemented in theENZO code, in order to avoid spurious temperature fluctuations dueto floating point round-off error when the kinetic energy is much

larger than the internal energy. We treat isotopes as passive scalarsthat are transported with the gas, and that decay with half-lives of2.62 Myr for 60Fe and 0.72 Myr for 26Al (Rugel et al. 2009; Norriset al. 1983).

The gas cools radiatively to 10 K using a 1D cooling curve createdfrom the CLOUDY package’s cooling table for metals and ENZO’snon-equilibrium cooling rates for atomic species of hydrogen andhelium (Abel et al. 1997; Ferland et al. 1998). This is implementedas tabulated cooling rates as a function of density and temperature(Jin et al. 2017). In addition to radiative cooling, the gas can alsobe heated via diffuse photoelectric heating in which electrons areejected from dust grains via far-ultraviolet (FUV) photons. Thisis implemented as a constant heating rate of 8.5 × 10−26 erg s−1

per hydrogen atom uniformly throughout the simulation box. Thisrate is chosen to match the expected heating rate assuming a UVbackground consistent with the Solar neighbourhood value (Draine2011). Self-gravity of the gas is also implemented.

We do not include dust grain physics because the typical drift ve-locity of the small dust (∼0.1μm) relative to gas at subparsec scalein the galactic disc is only 7.5 × 10−4 km s−1, much smaller thanthe typical turbulent velocity of the ISM (∼10 km s−1) (Wibking,Thompson & Krumholz 2018). Furthermore, analysis of Al and Fedust grains in SN ejecta constrains their sizes to be less than 0.01μm (Bocchio et al. 2016). Therefore, the dust grains and gas arevery well coupled at the spatial scale we resolve in this simulation.

2.2 Galaxy model

The galaxy is modelled in a 3D simulation box of (128 kpc)3 withisolated gravitational boundary conditions and periodic fluid bound-aries. The root grid is 1283 with an additional seven levels of refine-ment, producing a minimum cell size of 7.8125 pc. We refine a cellif the Jeans length, λJ = cs

√π/(Gρ), drops below 8 cell widths,

comfortably satisfying the Truelove et al. (1998) criterion. In ad-dition, to ensure that we resolve stellar feedback, we require thatany computational zone containing a star particle be refined to themaximum level. To keep the Jeans length resolved after collapsehas reached the maximum refinement level, we employ a pressurefloor such that the Jeans length is resolved by at least four cells onthe maximum refinement level. In addition to the static root grid, weimpose five additional levels of statically refined regions enclosingthe whole galactic disc of 14 kpc radius and 2 kpc height. Thisguarantees that the circular motion of the gas in the galactic disc iswell resolved, with a maximum cell size of 31.25 pc.

We use initial conditions identical to those of Tasker & Tan(2009). These are tuned to the Milky Way in its present state, but theGalaxy’s bulk properties were not substantially different when Solarsystem formed 4.567 Gyr ago (z ∼ 0.4). The simulated galaxy is setup as an isolated disc of gas orbiting in a static background potentialwhich represents both dark matter and a stellar disc component. Theform of the background potential is

�(r, z) = 1

2v2

c,0 ln

[1

r2c

(r2

c + r2 + z2

q2φ

)], (1)

where vc, 0 is the constant circular velocity at large radii, here setequal to 200 km s−1, r and z are the radial and vertical coordinates,the core radius is rc = 0.5 kpc, and the axial ratio of the potentialis qφ = 0.7. This corresponding circular velocity is

vc = vc,0r√r2

c + r2. (2)

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4028 Y. Fujimoto, M. R. Krumholz and S. Tachibana

The initial gas density distribution is

ρ(r, z) = κ√

c2s + σ 2

1D

2πGQzhsech2

(z

zh

), (3)

where κ is the epicyclic frequency, cs is the sound speed, here setequal to 6 km s−1, σ 1D is the 1D velocity dispersion of the gasmotions in the plane of the disc after the subtraction of the circularvelocity, Q is the Toomre stability parameter, and zh is the verticalscale height, which is assumed to vary with galactocentric radiusfollowing the observed radially dependent H I scale height for theMilky Way. Our disc is initialized with σ1D = 0.

The initial disc profile is divided radially into three parts. In ourmain region, between radii of r = 2 − 13 kpc, ρ is set so that Q = 1.The other regions of the galaxy, from 0 to 2 kpc and from 13 to14 kpc, are initialized with Q = 20. Beyond 14 kpc, the disc issurrounded by a static, very low density medium. We set the initialabundances of 60Fe and 26Al to 10−12, though this choice has nopractical effect since the initial abundances decay rapidly. In total,the initial gas mass is 8.6 × 109 M�, and the initial 60Fe and 26Almass are set to 8.6 × 10−3 M�.

Note that we do not include explicit spiral perturbations in ourgravitational potential, but that flocculent spiral structure none theless forms spontaneously in our simulation as a result of gas self-gravity (see Section 3.1). Similarly, we do not have a live modelof the stellar bulge, but we implicitly include its effects on the gasvia our potential, which has a bulge-like flattening at small radii.However, our simulation does not include the effects of a Galacticbar, nor does it include the effects of cosmological inflow or tidalinteractions with satellite galaxies. The influence of these effectsshould be addressed in a future work.

2.3 Star formation

Implementations of star formation in galaxy-scale simulations suchas ours are generally parametrized by two choices: a thresholddensity at which star formation begins, and an efficiency of star for-mation in cells above that threshold. In isolated galaxy simulationssuch as the one we perform, numerical experiments (e.g. Hopkins,Narayanan & Murray 2013a) have shown that observed galaxies arebest reproduced in simulations where the star formation thresholdis set based on criteria of gravitational boundedness, i.e. star for-mation should occur only in fluid elements that are gravitationallybound or nearly so at the highest available numerical resolution. Ina grid simulation such as ours, the criterion of boundedness is mostconveniently expressed in terms of the ratio of the local Jeans lengthλJ to the local cell size x. We set our star formation threshold suchthat gas is star forming if λJ/x < 4 for x at the maximum al-lowed refinement level (Truelove et al. 1997); note that this choiceguarantees that star formation occurs only in cells that have been re-fined to the highest allowed level. Rather than calculating the soundspeed on the fly, it is more convenient to note that, at the densitiesat which we will be applying this condition, the gas is always veryclose to the thermal equilibrium defined by equality between photo-electric heating and radiative cooling (Section 2.1). Consequently,we can reduce the condition for gas to be star forming to a simpleresolution-dependent density threshold by setting the sound speedbased on the equilibrium temperature as a function of density. Doingso and plugging in the various resolutions, we will use in this paper(see Section 3) yields number density thresholds for star formationof 12 cm−3 for a resolution x = 31 pc, 25.4 cm−3 for x = 15 pcand 57.5 cm−3 for x = 8 pc.

The second parameter in our star formation recipe characterizesthe star formation rate in (SFR) gas that exceeds the threshold. Weexpress the SFR density in cells that exceed the threshold as

dρ∗dt

= εffρ

tff. (4)

Here, ρ is the gas density of the cell, tff = √3π/32Gρ is the local

dynamical time, and εff is our rate parameter. Fortunately, the valueof εff is very well constrained by both observations and numericalexperiments. For observations, one can measure εff directly by avariety of methods, and the consensus result from most techniquesis that εff ≈ 0.01, with relatively little dispersion (e.g. Krumholz &Tan 2007; Krumholz, Dekel & McKee 2012; Evans, Heiderman &Vutisalchavakul 2014; Heyer et al. 2016; Vutisalchavakul, Evans &Heyer 2016; Leroy et al. 2017; Onus, Krumholz & Federrath 2018,though see Lee, Miville-Deschenes & Murray 2016 for a contrastingview). From the standpoint of numerical experiments, a number ofauthors have shown that only simulations that fix εff ≈ 0.01 yieldISM density distributions consistent with observational constraints(e.g. Hopkins et al. 2013b; Semenov, Kravtsov & Gnedin 2018).Given these constraints, we adopt εff = 0.01 for this work.

To avoid creating an extremely large number of star particleswhose mass is insufficient to have a well-sampled stellar popula-tion, we impose a minimum star particle mass, msf, and form starparticles stochastically rather than spawn particles in every cell ateach time-step. In this scheme, a cell forms a star particle of massmsf = 300 M� with probability

P =(

εffρ

tffx3t

)/msf, (5)

where x is the cell width, and t is the simulation time-step.In practice, all star particles in our simulation are created via thisstochastic method with masses equal to msf. Note that the choice ofthe star particle of mass 300 M� does not affect the total SFR inthe simulated galaxy as shown in fig. 1 in Goldbaum, Krumholz &Forbes (2015), and we show Appendix B that our star particles aresmall enough that we resolve the characteristic size scale on whichstar formation is clustered extremely well, so that our choice of starparticle mass does not affect the clustering of star formation either.Star particles are allowed to form in the main region of the discbetween 2 < r < 14 kpc.

2.4 Stellar feedback

Here, we describe a subgrid model for star formation feedbackthat includes the effects of ionizing radiation from young stars, themomentum and energy released by individual SN explosions, andgas and isotope injections from stellar winds and SNe. The inclusionof multiple forms of feedback is critical for producing results thatagree with observations in high-resolution simulations such as ours(e.g. Hopkins, Quataert & Murray 2011; Agertz et al. 2013; Stinsonet al. 2013; Renaud et al. 2013). In particular, simulations withenough resolution to capture the ≈5 Myr delay between the onset ofstar formation and the first SN explosions require non-SN feedbackin order to avoid overproducing stars (compared to what is observed)before SNe have time to disperse star-forming gas. We pause here tonote that this means that implementations of feedback are inevitablytuned to the resolution of the simulations being carried out, withsimulations that go to higher resolution requiring the inclusion ofmore physical processes to replace the artificial softening of gravitythat occurs at lower resolution. The feedback implementation we use

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Figure 1. The time evolution of SFR and isotope mass. The black solidline shows the total SFR in the galactic disc. The blue dotted and red dashedlines show the total mass of 60Fe and 26Al, respectively. The sharp featuresat 600 and 660 Myr are transients caused when we increase the resolution.

here is tuned to the ∼10 pc resolution we achieve, and is very similarto that of other authors who run simulations at similar resolution.

All star particles form with a uniform initial mass of 300 M�.Within each of these particles we expect there to be a few starsmassive enough to produce SN explosions. We model this usingthe SLUG stellar population synthesis code (da Silva, Fumagalli &Krumholz 2012; Krumholz et al. 2015). This stellar populationsynthesis method is used dynamically in our simulation; each starparticle spawns an individual SLUG simulation that stochasticallydraws individual stars from the IMF, tracks their mass- and age-dependent ionizing luminosities, determines when individual starsexplode as SNe, and calculates the resulting injection of 60Fe and26Al. In the SLUG calculations, we use a Chabrier IMF (Chabrier2005) with SLUG’s Poisson sampling option, Padova stellar evolu-tion tracks with Solar metallicity (Girardi et al. 2000), STARBURST99stellar atmospheres (Leitherer et al. 1999), and Solar metallicityyields from Sukhbold et al. (2016).

We include stellar feedback from photoionization and SNe, fol-lowing Goldbaum, Krumholz & Forbes (2016), though our numer-ical implementation is very similar to that used by a number ofprevious authors (e.g. Renaud et al. 2013). For the former, we usethe total ionizing luminosity S from each star particle calculated bySLUG to estimate the Stromgren volume Vs = S/αBn2, and comparewith the cell volume, Vc. Here, αB = 2.6 × 10−13 cm3 s−1 is thecase B recombination rate coefficient, n = ρ/μmH is the numberdensity, and μ = 1.27 and mH = 1.67 × 10−24 g are the meanparticle mass and the mass of an H nucleus, respectively. If Vs < Vc,the cell is heated to 104(Vs/Vc) K. If Vs > Vc, the cell is heated to atemperature of 104 K, and then we calculate the luminosity Sesc = S− αBn2Vc that escapes the cell. We distribute this luminosity evenlyover the neighbouring 26 cells, and repeat the procedure.

For SN feedback, a critical challenge in high-resolution simula-tions such as ours is that the Sedov–Taylor radius for SN remnantsmay or may not be resolved, depending on the ambient density inwhich the SN explodes. In this regime, several authors have carriedout numerical experiments showing that the feedback recipes thatbest reproduce the results of high-resolution simulations are thosethat switch smoothly injecting pure radial momentum in cases wherethe Sedov–Taylor radius is unresolved to adding pure thermal en-ergy in cases where it is resolved (e.g. Kimm et al. 2015; Hopkins

et al. 2018). Our scheme, which is identical to that used in Gold-baum et al. (2016), is motivated by this consideration. We identifyparticles that will produce SNe in any given time-step. For eachSN that occurs, we add a total momentum of 3 × 105 M� km s−1,directed radially outward in the 26 neighbouring cells. This momen-tum budget is consistent with the expected deposition from singleSNe (Gentry et al. 2017). The total net increase in kinetic energyin the cells surrounding the SN host cell is then deducted from theavailable budget of 1051 erg and the balance of the energy is thendeposited in the SN host cell as thermal energy. This scheme meetsthe requirement of smoothly switching from momentum to energyinjection depending on the ambient density: if the explosion occursin an already-evacuated region such that the gas density is low, thekinetic energy added in the process of depositing the radially out-ward momentum will be �1051 erg, and the bulk of the SN energywill be injected as pure thermal energy. In a dense region, on theother hand, little thermal energy will remain, and only the radialmomentum deposited will matter. In the higher resolution phasesof the simulation (x = 15 and 8 pc), we increase the momentumbudget to 5 × 105 M� km s−1 in order to maintain approximatelythe same total SFR; given that the actual momentum budget is un-certain by a factor of ≈10 due to the effects of clustering (Gentryet al. 2017), this value is still well within the physically plausiblerange.

We include gas mass injection from stellar winds and SNe toeach star particle’s host cell each time-step. The mass-loss rate ofeach star particles is calculated from the SLUG stellar populationsynthesis. Note that we do not include energy injection from stellarwinds; these will be included in future work. However, even thoughthe simulation does not include the effect, the total SFR in thesimulated galaxy is consistent with observations.

We include isotope injection from stellar winds and SNe, which iscalculated from the mass-dependent yield tables of Sukhbold et al.(2016). The explosion model for massive stars is 1D, of a singlemetallicity (solar) and does not include any effects of stellar rota-tion. The chemical yields are deposited to the host cell. As discussedin Sukhbold et al. (2016), their nucleosynthesis model overpredicts2

the 60Fe to 26Al compared to that determined from γ -ray line ob-servations (Wang et al. 2007). They note that the discrepancy mighthave to do with errors in poorly known nuclear reaction rates, es-pecially for 26Al(n, p)26Mg, 26Al(n, α)23Na, 59, 60Fe(n, γ )60, 61Fe, orwith uncertainties in stellar mixing parameters such as the strengthof convective overshoot. Rotational mixing is another possible effectthat is not considered in their chemical yields (Chieffi & Limongi2013; Limongi & Chieffi 2018). To ensure that our 60Fe/26Al ratio isconsistent with observations, we modify their tables slightly by re-ducing the 60Fe yield by a factor of five and doubling the 26Al yield.This brings our galaxy-averaged ratios of 60Fe/26Al, 60Fe/SFR, and26Al/SFR into good agreement with observations. Although uncer-tainties in the chemical yields might affects our results, we expectthe effect to be at most a factor of ten, not orders of magnitude,since this is the current level of discrepancy between the numerical

2Sukhbold et al. (2016) compared their ejected mass ratio of 60Fe/26Al (=0.9) with the observed steady-state mass ratio of 0.34 (Wang et al. 2007), andstated that their yield should be corrected by a factor of three. However, asteady-state mass ratio should be used, not the ejected mass ratio, to comparewith the observed mass ratio. The steady-state mass ratio can be obtainedby multiplying the ratio of half-lives, as 0.9 × (2.62 Myr/0.72 Myr) = 3.3.This steady-state mass is 10 times larger than the observed steady-state massratio. That is why we modify their tables by reducing the 60Fe yield by afactor of five and doubling the 26Al yield.

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Figure 2. The morphology of the galactic disc. Panels show the gas (left), 60Fe (middle), and 26Al (right) surface densities of the face-on disc at t = 750 Myr.Each image is 28 kpc across. The galactic disc rotates anticlockwise. The two circles indicate Galactocentric radii of 7.5 kpc and 8.5 kpc, roughly boundingthe Solar annulus.

Figure 3. Same as Fig. 2, but zoomed-in on a spot near the Solar Circle. Panels show the gas (top left), SFR (top right), 60Fe (bottom left), and 26Al (bottomright) surface densities at t = 750 Myr. The two arcs show Galactocentric radii of 7.5 and 8.5 kpc, bounding the Solar annulus.

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Figure 4. Same as Fig. 3, but showing hot gas (>106K) on the top right panel.

results and the observations. It would be worthwhile repeating oursimulations in the future with other models of chemical yields (Ek-strom et al. 2012; Limongi & Chieffi 2006, 2018; Chieffi & Limongi2013; Nomoto et al. 2006; Nomoto, Kobayashi & Tominaga 2013;Pignatari et al. 2016).

3 SIMULATION R ESULTS

3.1 Evolution of the disc

To determine the equilibrium distributions of isotopes in newlyformed stars, we use a relaxation strategy to allow the simulatedgalaxy to settle into statistical equilibrium at high resolution. Wefirst run the simulation at a resolution of 31 pc for 600 Myr, corre-sponding to two rotation periods at 10 kpc from the galactic centre.This time is sufficient to allow the disc to settle into statistical steadystate, as we illustrate in Fig. 1, which shows the time evolution of thetotal SFR and total 60Fe and 26Al masses within the galaxy. We thenincrease the resolution from 31 to 15 pc and allow the disc to settleback to steady state at the new resolution, which takes until 660 Myr.At that point, we increase the resolution again, to 8 pc. These re-

finement steps are visible in Fig. 1 as sudden dips in the SFR, whichoccur because it takes some time after we increase the resolutionfor gas to collapse past the new, higher star formation threshold,followed by sudden bursts as a large mass of gas simultaneouslyreaches the threshold. However, feedback then pushes the systemback into equilibrium. In the equilibrium state the SFR is 1−3 M�yr−1, consistent with the observed Milky-Way SFR (Chomiuk &Povich 2011). Similarly, the total SLR masses in the equilibriumstate are 0.7 M� for 60Fe and 2.1 M� for 26Al, respectively, consis-tent with masses determined from γ -ray observations (Diehl 2017;Wang et al. 2007). Note that, as we change the resolution, the steady-state SFR and SLR abundances vary at the factor of ≈2 level. Thisis not surprising, because our stellar feedback model operates on astencil of 33 cells around each star particles, and thus volume overwhich we inject feedback varies as does the resolution. However, wenote that the variations in equilibrium SFR and SLR mass with res-olution are well within both the observational uncertainties on thesequantities.

Fig. 2 shows the global distributions of gas and isotopes in thegalactic disc at t = 750 Myr, when the maximum resolution is 8 pcand the galactic disc is in a quasi-equilibrium state. Fig. 3 shows

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Figure 5. The abundance ratios of short-lived isotopes in newly formedstars. The central panel shows the joint PDF of 60Fe/56Fe and 26Al/27Alfrom our simulations, with colours showing probability density and blackpoints showing individual stars in sparse regions. The top and right-hand pan-els show the PDFs of 60Fe/56Fe and 26Al/27Al individually, with simulationsshown in blue. All simulation data are for stars formed from 740 to 750 Myr,at Galactocentric radii from 7.5 to 8.5kpc. Green bands show the uncertaintyrange of Solar system meteoritic abundances (Lee et al. 1976; Mishra &Goswami 2014; Tang & Dauphas 2015; Telus et al. 2018); for 60Fe, dueto the wide range of values reported in the literature, we also show threerepresentative individual measurements as indicated in the legend.

the same data, zoomed-in on a 3.5 kpc region centred on the SolarCircle.3 The figures show that the disc is fully fragmented, andhas produced GMCs and star-forming regions. The distributions of60Fe and 26Al are strongly correlated with the star-forming regions,which correspond to the highest density regions (reddish colours)visible in the gas plot. This is as expected, since these isotopes areproduced by massive stars, which, due to their short lives, do nothave time to wander far from their birth sites.

However, there are important morphological differences betweenthe distributions of 60Fe, 26Al, and star formation. The 60Fe distribu-tion is the most extended, with the typical region of 60Fe enrichmentexceeding 1 kpc in size, compared to ∼100 pc or less for the den-sity peaks that represent star-forming regions. The 26Al distributionis intermediate, with enriched regions typically hundreds of pc inscale. The larger extent of 60Fe compared to 26Al is due to its largerlifetime (2.62 Myr versus 0.72 Myr for 26Al) and its origin solelyin fast-moving SN ejecta (as opposed to pre-SN winds, which con-tribute significantly to 26Al).

In addition to the comparison between SLRs and star formation,it is interesting to compare SLRs to the distribution of hot gasproduced by SNe (defined here as gas with temperature T > 106

K), which we show in Fig. 4. We see that, as expected, regions of60Fe and 26Al enrichment correlate well with bubbles of hot gas.However, it is interesting to note that the outer edges of the 60Feor 26Al bubbles seen in Fig. 4 extend significantly further than the

3Simulation movies are available at https://sites.google.com/site/yusuke777fujimoto/data

bubbles of hot ISM. This could be a result either of cooling ofthe hot gas on time-scales shorter than the decay of SLRs, or ofrapid mixing of SLRs into cooler regions. Regardless, our findingthat regions of SLR enrichment are generally larger in extent thanregions of hot gas may be testable in the future as higher resolutionobservations of γ -ray emission from SLRs observed in situ in theISM become available.

3.2 Abundance ratios in newborn stars

To investigate abundance ratios of isotopes in newborn stars, when-ever a star particle forms in our simulations, we record the abun-dances of 60Fe and 26Al in the gas from which it forms, since theseshould be inherited by the resulting stars. We do not add any ad-ditional decay, because our stochastic star formation prescriptiondoes not immediately convert gas to stars as soon as it crossesthe density threshold, and instead accounts for the finite delay be-tween gravitational instability and final collapse. Fig. 5 shows theprobability distribution functions (PDFs) for the abundance ratios60Fe/56Fe and 26Al/27Al; we derive the masses of the stable isotopes56Fe and 27Al from the observed abundances of those species inthe Sun (Asplund et al. 2009), and we measure the PDFs for starparticles that form between 740 and 750 Myr in the simulation, atgalactocentric radii from 7.5 to 8.5 kpc (i.e. within ≈0.5 kpc ofthe Solar Circle). However, the results do not strongly vary withgalactocentric radius, as shown in Appendix A. We also show thatthe PDFs are converged with respect to spatial resolution at theirhigh-abundance ends (though not on their low-abundance tails) inAppendix B.

In Fig. 5 we also show meteoritic estimates for these abundanceratios (Lee et al. 1976; Mishra & Goswami 2014; Tang & Dauphas2015; Telus et al. 2018). The PDF of 60Fe peaks near 60Fe/56Fe∼ 3 × 10−7, but is ∼2 orders of magnitude wide, placing all themeteoritic estimates well within the ranges covered by the simulatedPDF. The 26Al abundance distribution is similarly broad, but themeasured meteoritic value sits very close to its peak, as 26Al/27Al∼ 5 × 10−5. Clearly, the abundance ratios measured in meteoritesare fairly typical of what one would expect for stars born near theSolar Circle, and thus the Sun is not atypical.

4 D ISCUSSION

Our simulations suggest a mechanism by which the SLRs came to bein the primitive Solar system that is quite different than proposed inearlier work based on smaller scale simulations or analytic models.We call this new contamination scenario ‘inheritance from Galactic-scale correlated star formation’. Our scenario differs substantiallyfrom the triggered collapse or direct injection scenarios in that bothof these require unusual circumstances – the core that forms theSun is either at just the right distance from an SN to be triggeredinto collapse but well mixed, or the protoplanetary disc was hit bySN ejecta and managed to capture them without being destroyed. Ineither case stars with SLR abundances like those of the Solar systemshould be rare outliers, while we find that the Sun’s abundances aretypical.

However, the scenario illustrated in our simulations is also verydifferent from the GMC confinement hypothesis. To see why, oneneed only examine Fig. 3. Observed GMCs, and those in our simu-lations, are at most ∼100 pc in size, whereas in Fig. 3 we clearly seethat regions of 60Fe and 26Al contamination are an order of magni-tude larger. This difference between our simulations and the GMCconfinement hypothesis is also visible in the distribution of 26Al on

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Figure 6. Distributions of gas, star formation, and SLRs in Galactic coordinates, as viewed from the position of the Sun (i.e. a point 8 kpc from the Galacticcentre). Panels show the gas, SFR, 60Fe and 26Al distributions (from top to bottom) in Galactic coordinates. Note that, although the absolute scales on thecolour bars in each panel differ, all panels use the same dynamic range, and thus the distributions are directly comparable. The scalloping pattern that is visibleat high latitudes and toward the outer galaxy is an artefact due to aliasing between the Cartesian grid and the angular coordinates in regions where the resolutionis low.

the sky as seen from Earth. Fig. 6 shows all-sky maps of the gas,SFR, 60Fe, and 26Al as viewed from a point 8 kpc from the GalacticCentre (i.e. at the location of the Sun). We should not regard Fig. 6as an exact prediction of the γ -ray sky as seen from Earth, since wehave not taken care to replicate the Sun’s placement relative to spiralarms, nor have we tried to match the sky positions of local structuressuch as the Sco-Cen association that may have a large impact onwhat we observe from Earth. However, it is none the less interestingto examine the large-scale qualitative behaviour of the map shownin Fig. 6, and its implications. If SLRs are confined by GMCs, thenγ -rays from 26Al decay should have an angular thickness on thesky comparable to that of star-forming regions. Fig. 6 clearly showsthat this is not the case in our simulations: 60Fe and 26Al extend toGalactic latitude b = 4◦−5◦, while star-forming regions are con-fined to b < 2◦. The difference in scale heights we find is consistentwith observations. The Galactic CO survey of Dame, Hartmann &Thaddeus (2001) finds that most emission is confined to Galacticlatitudes b < 2◦, while the γ -ray emission maps of 26Al (Pluschkeet al. 2001; Bouchet et al. 2015) show a thick disc with b ≈ 5◦. Our

simulation successfully reproduces the observed difference in 26Aland CO angular distribution.

We can make this discussion more quantitative by examiningthe distribution of 60Fe and 26Al and their correlation with the gasand star formation properties of the galaxy. We first examine thedistribution of the SLRs with respect to gas density and temperature,as illustrated in Fig. 7. We find that only 30 per cent of the 60Feand 56 per cent of the 26Al by mass are found in GMCs (defined asgas with a density above 100 H cm−3), compared to a total GMCmass fraction of 16 per cent; thus 60Fe is overabundant in GMCscompared to the bulk of the ISM by less than a factor of 2, and26Al by less than a factor of 3.5. These modest enhancements areinconsistent with the hypothesis that SLRs abundances are highin the Solar system because SLRs are trapped within long-livedGMCs.

We can also reach a similar conclusion by examining the spatialcorrelation of star formation with SLRs. For any 2D fields f (r) andg(r) defined as a function of position r within the galactic disc, wecan define the normalized spatial cross-correlation function (f∗g)(r)

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Figure 7. Mass distributions with respect to gas temperature versus density. Left is the gas, middle is 60Fe, and right is 26Al at t = 750 Myr.

Figure 8. Normalized spatial cross-correlations (f∗g)(r) between the SFRsurface density and the surface densities of 60Fe and 26Al divided by the gassurface density.

as

(f ∗ g)(r) =⟨∫

f (r ′)g(r ′ − r) dr ′⟩∫f (r ′)g(r ′) dr ′ (6)

where r = |r|, and the angle brackets indicate an average over allpossible angles of the displacement vector r . In practice, we cancompute the correlation numerically using projected images suchas those shown in Fig. 2 for two quantities f and g. The denominatoris simply the product of the two images, while we can obtain theintegral in the numerator for a displacement vector r by shifting oneof the images by r , multiplying the shifted and unshifted images, andmeasuring product of the two images. We then compute the averageover angle by averaging the numerator over shifts of the samemagnitude r = |r|. We show the spatial cross-correlation betweenstar formation and element abundance ratios in Fig. 8. As one can seefrom the figure, star formation is correlated with 60Fe abundance onscales of 1 kpc and 26Al abundance on scales of hundreds of parsec,much larger than an individual GMC or star-forming complex. Thedifference of the correlation scales between the 60Fe and 26Al comesfrom the different lifetimes (2.62 Myr versus 0.72 Myr) and thefact that 60Fe is added to the ISM only through fast-moving SN

ejecta, while 26Al has contributions from both SNe and pre-SNstellar winds. This is consistent with the different morphologicaldistributions of 60Fe and 26Al as shown in Fig. 3. The results do notstrongly vary with galactocentric radius, as shown in Appendix A.

The overall picture that emerges from our simulations is thatSLR abundances in newborn stars are large because star formationis highly correlated in time and space (Efremov & Elmegreen 1998;Gouliermis et al. 2010, 2015, 2017; Grasha et al. 2017a,b). SNejecta are not confined to individual molecular clouds, and insteaddeposit radioactive isotopes in the atomic gas over ∼1 kpc fromtheir parent molecular clouds. However, because star formation iscorrelated, and because molecular clouds are not closed boxes butinstead continually accrete the atomic gas during their star-forminglives (Fukui & Kawamura 2010; Goldbaum et al. 2011; Zamora-Aviles, Vazquez-Semadeni & Colın 2012), the pre-enriched atomicgas within ∼1 kpc of a molecular cloud stands a far higher chanceof being incorporated into a molecular cloud and thence into starswithin a few Myr than does a random portion of the ISM at similardensity and temperature. Conversely, the atomic gas in a galaxy thatwill be incorporated into a star a few Myr in the future does notrepresent an unbiased sampling of all the atomic gas in the galaxy.Instead, it is preferentially that atomic gas that is close to sites ofcurrent star formation, and thus is far more likely than average tohave been contaminated with SLRs. It is the Galactic-scale cor-relation of star formation that is the key physical mechanism thatproduces high SLR abundances in the primitive Solar system andother young stars.

5 C O N C L U S I O N S

SLRs such as 60Fe and 26Al are radioactive elements with half-livesless than 15 Myr that studies of meteorites have shown to be presentat the time when the most primitive Solar system bodies condensed.The most likely origin site for the 60Fe and 26Al in meteorites isnucleosynthesis in massive stars, but the exact delivery mechanismby which these elements entered the Solar system’s protoplanetarydisc are still debated.

To address this question, we have performed the first chemohy-drodynamical simulation of the entire Milky-Way Galaxy (Fig. 2),including stochastic star formation and stellar feedback in the formof H II regions, SNe, and element injection. Our simulations have

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enough resolution to capture individual SNe, so that we can prop-erly measure the full range of variation in SLR abundances thatresults from the stochastic nature of element production and trans-port. From our simulations, we measure the expected distributionof 60Fe/56Fe and 26Al/27Al ratios for all stars in the galaxy (Fig. 5).We find that the Solar abundance ratios inferred from meteorites arewell within the normal range for Milky-Way stars; contrary to somemodels for the origins of SLRs, the Sun’s SLR abundances are notatypical.

Our results lead us to propose a new enrichment scenario: SLRenrichment via Galactic-scale correlated star formation. We findthat GMCs are at most 100 pc in size and their star-forming regionsare much smaller, while regions of 60Fe and 26Al contaminationdue to SNe are an order of magnitude larger (Fig. 3). The extremelybroad distribution of 26Al produced in our simulations is consistentwith the observed distribution on the sky, which shows an angularscale height that is close to twice that of the molecular gas and starformation in the Milky Way (Fig. 6). The SLRs are not confinedto the molecular clouds in which they are born (Fig. 7). However,SLRs are none the less abundant in newborn stars because starformation is correlated on galactic scales (Fig. 8). Thus, althoughSLRs are not confined, they are in effect pre-enriching a halo of theatomic gas around existing GMCs that is very likely to be subse-quently accreted or to form another GMC, so that new generationsof stars preferentially form in patches of the galaxy contaminatedby previous generations of stellar winds and SNe.

In future work, we will extend our simulations to include otherSLRs such as 41Ca and 53Mn, which also have been claimed to placesevere constraints on the birth environment of the Solar system(Huss et al. 2009).

AC K N OW L E D G E M E N T S

The authors would like to thank the referee, Roland Diehl, for hiscareful reading and helpful suggestions. Simulations were carriedout on the Cray XC30 at the Center for Computational Astrophysicsof the National Astronomical Observatory of Japan and the NationalComputational Infrastructure, which is supported by the AustralianGovernment. YF and MRK acknowledge support from the Aus-tralian Government through the Australian Research Council’s Dis-covery Projects funding scheme (project DP160100695). Compu-tations described in this work were performed using the publiclyavailable ENZO code (Bryan et al. 2014; http://enzo-project.org),which is the product of a collaborative effort of many indepen-dent scientists from numerous institutions around the world. Theircommitment to open science has helped make this work possi-ble. We acknowledge extensive use of the YT package (Turk et al.2011; http://yt-project.org) in analysing these results and the authorswould like to thank the YT development team for their generous help.

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A P P E N D I X A : R A D I A L D E P E N D E N C E

To determine whether our stellar abundance distributions are typical in the whole galaxy, we examine the distributions for stars formed in1 kpc-wide annuli centred on galactocentric radii from 4 to 10 kpc. We show these radially resolved distributions in Fig. A1. The distributions

Figure A1. Dependence of the 60Fe/56Fe and 26Al/27Al PDFs on Galactocentric radius. The histograms are each measured for stars formed within a1 kpc-wide annulus centred at the Galactocentric radius indicated in the legend, at t = 740−750Myr.

clearly do not strongly vary with galactocentric radius. That means that most planetary system in the Galaxy could come to have the highabundance ratios of 60Fe/56Fe and 26Al/27Al, and therefore the birth environment of the Solar system is not atypical not only near the SolarCircle but also for a broad region in the Milky Way.

To determine if the physical explanation for these PDFs is the same at all galactocentric radii, in Fig. A2 we show the spatial correlation

Figure A2. Dependence of spatial cross-correlation functions between star formation and element abundance ratios on Galactocentric radius.

(equation 6) between star formation and SLR abundance measured at different galactocentric radii; we compute these functions using the sameprocedure as described in Section 4, except that we set the values of all pixels outside the target annulus to zero, so they do not contribute tothe correlation. Although there is clearly some scatter in correlation with radius, the qualitative result that 60Fe correlates with star formationon scales of several hundred pc, and 26Al on scales of ∼100 pc, appears to be the same at all galactocentric radii. This strongly suggests thatthe correlation is a result of the physics of stellar feedback and the lifetimes of the SLRs, rather than on any particular characteristic of thestar-forming environment.

A P P E N D I X B: R E S O L U T I O N A N D C O N V E R G E N C E

Because we find that the clustering of star formation is crucial to our results, it is important to demonstrate that the amount of clustering in oursimulations is not artificially enhanced by our choice of star particle mass, since by construction stars that form within a single star particleare perfectly correlated. To investigate this possibility, we must verify that the true clustering scale of star formation in our simulations ismuch larger than the size of a single one of our star particles. We therefore calculate the two-point correlation function of star particles withages <1 Myr, ξ (r), which traces the amplitude of clustering of star particles as a function of scale. We perform this calculation using the

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4038 Y. Fujimoto, M. R. Krumholz and S. Tachibana

Figure B1. Two-point correlation function of star particles, ξ (R), which traces the amplitude of clustering of star particles as a function of scale. The arrowshows the size scale of star particle, defined as (msf/ρsf)1/3, where msf (=300 M�) is star particle mass and ρsf (=57.5mp cm−3) is the threshold density forstar formation.

Figure B2. Resolution study for SLR abundance PDFs in newly formed stars at Galactocentric radii from 7.5 to 8.5kpc. The left-hand panel is 60Fe/56Fe,and right-hand panel is for 26Al/27Al. The black dash–dotted line shows the distribution of abundances at 31 pc resolution run (t = 590−600Myr), the browndashed line is 15 pc resolution (t = 650−660) Myr, and the tan solid line is at 8 pc resolution (t = 740−750Myr).

clustering estimator of Davis & Peebles (1983),

ξ (r) = nR

nD

DD

DR− 1. (B1)

Here, DD is the number of star particle pairs with a separation in the range r ± r (r = 5 pc) computed using the positions of stars outputby our simulations (i.e. the ‘data’ catalogue, D), while DR is the same quantity computed using pairs of particles where one is drawn from theactual list of stars (D), and the other is drawn from a ‘random’ catalogue (R) generated by randomly placing stars in the same volume as D; nD

and nR are the mean number densities of star particles in the data and random catalogues, respectively. For the purposes of our computation,we take our data catalogue to be the set of all star particles younger than 1 Myr at our final output time within a cubical region 2 kpc on aside, centred on the Solar circle; the region we use is the same one shown in Fig. 3. For our random catalogue, we use 1000 times as manyrandom star particles as in the data catalogue.

We show the result of this computation in Fig. B1. We can see from the figure that the characteristic size scale on which star formationis clustered in our simulations is ≈40–50 pc. For comparison, the size scale of ISM sampled by an individual star particle is �(msf/ρsf)1/3,where msf (=300 M�) is star particle mass and ρsf (=57.5mp cm−3 for mean particle mass mp) is the threshold density for star formation;this is an inequality because gas does not form stars immediately upon reaching ρsf, but may in fact collapse to somewhat higher density andsmaller size scale before doing so. Our upper limit on the characteristic size of a star particle is 5.5 pc, which is shown in Fig. B1 as an arrow.Thus, the size scales of stellar clustering in our simulation are roughly an order of magnitude larger than the sizes of individual star particles,and thus the choice of star particle size does not influence the degree of clustering.

To determine whether our stellar abundance distributions are converged, we compare the distributions we measure for stars formed at740–750 Myr of evolution, when our resolution is 8 pc at the galaxy has reached steady state, to those formed at 590–600 Myr (steadystate at 31 pc resolution) and 650–660 Myr (steady state at 15 pc resolution). We show the results in Fig. B2. We find that, although thepeaks of the PDFs move to higher values with higher resolution, the high-end tails converge to 10−6–10−5 for 60Fe/56Fe, and 10−4–10−3 for

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SLRs from Galactic-scale star formation 4039

26Al/27Al. Thus, we are well converged on the upper half of the abundance distribution. Moreover, given the broad range of uncertainties inthe meteoritic abundance, the shifts we do see with resolution do not change the qualitative conclusion that Sun’s SLR abundances are withinthe normal range expected for Milky-Way stars.

This paper has been typeset from a TEX/LATEX file prepared by the author.

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